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Topic 6 Real and Complex Number Systems II 9.1 – 9.5, 12.1 – 12.2 Algebraic representation of complex numbers including: • Cartesian, trigonometric (mod-arg) and polar form • definition of complex numbers including standard and trigonometric form • geometric representation of complex numbers including Argand diagrams • powers of complex numbers • operations with complex numbers including addition, subtraction, scalar multiplication, multiplication and conjugation Topic 6 Real and Complex Number Systems II 2 i Definition = -1 i = -1 A complex number has the form z = a + bi (standard form) where a and b are real numbers We say that Re(z) = a and that Im(z) = b [the real part of z] [the imaginary part of z] i =i i2 = -1 i3 = -i i4 = 1 i5 = i i6 = -1 i7 = -i i8 = 1 Question : What is the value of i2003 ? Model : Solve x 2 x 2 0 2 b b 4ac x 2a 2 2 2 4 21 2 2 4 2 2 2i 2 1 i 2 Equality If a + bi = c + di then a = c and b = d Addition a+bi + c+di = (a+c) + (b+d)i e.g. 3+4i + 2+6i = 5+10i e.g. 2+6i – (4-5i) = 2+6i-4+5i = -2+11i Scalar Multiplication 3(4+2i) = 12+6i Multiplication (3+4i)(2+5i) = 6+8i+15i+20i2 = 6 + 23i + -20 = -14 + 23i (2+3i)(4-5i) = 8-10i+12i-15i2 = 8 + 2i -15 i2 = 23 + 2i In general (a+bi)(c+di) = (ac-bd) + (ad+bc)i Exercise Exercise NewQ P 227, 234 Exercise 9.1, 9.3 FM P 168 Exercise 12.1 Determine the nature of the roots of each of the following quadratics: (a) x2 – 6x + 9 = 0 (b) x2 + 7x + 6 = 0 (c) x2 + 4x + 2 = 0 (d) x2 + 4x + 8 = 0 Determine the nature of the roots of each of the following quadratics: (a) x2 – 6x + 9 = 0 (b) x2 + 7x + 6 = 0 (c) x2 + 4x + 2 = 0 (d) x2 + 4x + 8 = 0 (a) x2 – 6x + 9 = 0 = 36 – 4x1x9 =0 ∴ The roots are real and equal [x=3] Determine the nature of the roots of each of the following quadratics: (a) x2 – 6x + 9 = 0 (b) x2 + 7x + 6 = 0 (c) x2 + 4x + 2 = 0 (d) x2 + 4x + 8 = 0 (b) x2 + 7x + 6 = 0 = 49 – 4x1x6 = 25 ∴ The roots are real and unequal [ x = -1 or -6 ] Determine the nature of the roots of each of the following quadratics: (a) x2 – 6x + 9 = 0 (b) x2 + 7x + 6 = 0 (c) x2 + 4x + 2 = 0 (d) x2 + 4x + 8 = 0 (c) x2 + 4x + 2 = 0 = 16 – 4x1x2 = 8 ∴ The roots are real, unequal and irrational [ x = -2 2 ] Determine the nature of the roots of each of the following quadratics: (a) x2 – 6x + 9 = 0 (b) x2 + 7x + 6 = 0 (c) x2 + 4x + 2 = 0 (d) x2 + 4x + 8 = 0 (d) x2 + 4x + 8 = 0 = 16 – 4x1x8 = -16 ∴ The roots are complex and unequal [ x = -2 4i ] Exercise FM P 232 Exercise 9.2 Division of complex numbers 3 2i Model 4 3i 3 2i 4 3i 4 3i 4 3i 12 9i 8i 6i 2 16 9 6 17 i 25 6 17 i 25 25 Try this on your GC Exercise NewQ P 239 Exercise 9.4 Exercise • Prove that the set of complex numbers under addition forms a group • Prove that the set of complex numbers under multiplication forms a group Model : Show that the set {1,-1,i,-i} forms a group under multiplication x 1 -1 i -i 1 1 -1 i -i -1 -1 1 -i i i i -i -1 1 -i -i i 1 -1 • Since every row and column contains every element , it must be a group Exercise NewQ P 245 Exercise 9.5 Argand Diagrams Model : Represent the complex number 3+2i on an Argand diagram or Model : Show the addition of 4+i and 1+2i on an Argand diagram 4 y 2 x 0 -6 -4 -2 0 -2 -4 2 4 6 Draw the 2 lines representing these numbers 4 y 2 x 0 -6 -4 -2 0 -2 -4 2 4 6 Complete the parallelogram and draw in the diagonal. This is the line representing the sum of the two numbers 4 y 2 x 0 -6 -4 -2 0 -2 -4 2 4 6 Exercise New Q P300 Ex 12.1 Model : Express z=8+2i in mod-arg form y 4 (8,2) 2 x 0 -10 -8 -6 -4 -2 0 -2 -4 2 4 6 8 10 Model : Express z=8+2i in mod-arg form x r x r cos y r y r sin cos sin 8 2i x iy r cos r i sin r cos i sin r cis y 4 (8,2) 2 r 0 -10 -8 -6 -4 -2 0 -2 -4 y 2 x 4 x 6 8 10 Model : Express z=8+2i in mod-arg form r 82 22 68 tan 82 14o y z 8 2i 68 cis 14o 4 (8,2) 2 r 0 -10 -8 -6 -4 -2 0 -2 -4 2 x 4 6 8 10 Model : Express 4 3i 4 in mod arg form 3i 4 3i 3i 3i 4( 3 i ) 31 y 3 i r cis 3i 3 r ( 3 ) 2 12 2 2 tan 1 1 3 r 6 r cis 2 cis 6 0 -1 0 θ x 1 2 3 Model: Express 3 cis /3 in standard form 3 cis 3 3(cos 3 i sin 3 ) 3( i 3 2 1 2 3 2 3 3 2 i ) Exercise New Q P306 Ex 12.2